# Mechanical-Analytical Soil-Dependent Fragility Curves of Existing RC Frames with Column-Driven Failures

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## Abstract

**:**

## 1. Introduction

## 2. Determination of Displacement Capacity of 2D Frame

#### 2.1. Simplified Analytical Model of A 2D Frame

#### 2.2. Frame Section Capacity at x = D, L, C

#### 2.3. Column Capacity at $x=D,\text{}L,\text{}C$

#### 2.4. Story Capacity at $x=D,\text{}L,\text{}C$

#### 2.5. Frame Capacity at x = D, L, C

## 3. Determination of Displacement Demand on a 2D Frame

#### 3.1. Simplified Modal Analysis

#### 3.2. Equivalent SDOF System

#### 3.3. Bilinearization

#### 3.4. Displacement Demands at x = D, L, C

## 4. Development of Fragility Curves for Frame Typologies

#### 4.1. Definition of Fragility Curve

#### 4.2. Selection of Frame Typologies

#### 4.3. Effects of Soil Class and Location

#### 4.4. Monte Carlo Analyses

#### 4.5. Resulting Fragility Curves

#### 4.6. Comparison with Literature Fragility Curves

## 5. Conclusions

**Location and soil class influence:**when developing analytical fragility curves, the influence of the local hazard curve and of the local soil class must be considered. FCs pertaining to the same typology/building change when used at different locations and/or on different soil classes. This induces significant errors on risk and scenario studies at the territorial level. To carry out this study in a more effective manner, a strategy is under development aiming at transforming, through analytical closed-form functions, FCs developed on a certain location and soil class to another one. This is beyond the scope of the present study and will be discussed and presented in a future article.**Construction age:**RC frames fragility is significantly dependent on the construction age when it spans from pre-seismic-code to seismic-code periods. The fragility curves of the two epochs provide insightful information about the vulnerability features of the structures with respect to their construction age.**Building height:**the building height/number of stories is a crucial parameter to the evaluation of the fragility curves, since the LS-exceedance probability increases with the height/number of stories (here, studied only up to five). Thus, it is effectively used as a key parameter to define different building typologies.

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Conflicts of Interest

## References

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**Figure 1.**Moment-curvature relationship of a column section with the corresponding Limit State criteria for Damage limitation Limit State (DLS, tension steel yielding), Life safety Limit State (LLS, concrete cover crushing), and Collapse prevention Limit State (CLS, concrete core crushing).

**Figure 4.**Geometry of the frame typologies. Number of storeys ranges from 1 to 5, while bay spans ${L}_{i}$ are given in Table 3.

**Figure 12.**Comparison between the soil-aggregated FCs and some fragility curves from other authors for: (

**a**) DLS; (

**b**) LLS; and (

**c**) CLS.

**Table 1.**Flexural failure: equations for resisting moment and curvature at three LSs for the case of rectangular sections with symmetric reinforcement.

Damage Limitation Limit State (DLS) | |

Moment capacity | ${M}_{DLS}=\frac{{\epsilon}_{yd}^{2}}{{\epsilon}_{cu}^{2}}\left[0.45{n}_{S}\left(0.8-{n}_{S}\right)+2.4\left({\omega}_{s}+0.015\right)\right]b{d}^{2}{f}_{c}$ |

Curvature capacity | ${\varphi}_{DLS}=\frac{{\epsilon}_{y}}{2d}\left(3+\sqrt{{n}_{S}}\right)$ |

Life Safety Limit State (LLS) | |

Moment capacity | ${M}_{LLS}=\left[{\omega}_{s}+\frac{1}{2}{n}_{S}\left(1-\frac{{n}_{S}}{{\eta}_{f}}\right)\right]b{d}^{2}{f}_{c}$ |

Curvature capacity | ${\varphi}_{LLS}=\frac{{\epsilon}_{cu}}{d}\frac{0.8}{{n}_{S}}{\eta}_{f}$ |

Collapse Prevention Limit State (CLS) | |

Moment capacity | ${M}_{CLS}=\left[{\omega}_{s}+\frac{1}{2}{n}_{S}\left(1-\frac{{n}_{S}}{{\eta}_{f}}\right)\right]b{d}_{c}^{2}{f}_{c}$ |

Curvature capacity | ${\varphi}_{CLS}={\varphi}_{LLS}\frac{d}{{d}_{c}}\left(1+\frac{1}{4}\frac{{\tilde{\sigma}}_{2}}{{\epsilon}_{cu}}\right)$ |

HRC | HAZUS99 | Vision2000 | ATC-13 | EMS98 | Limit State | |
---|---|---|---|---|---|---|

Grade | Damage State | |||||

Slight | Slight damage | Fully operational | Slight | Grade 1 | ||

Light | Operational | Light | Grade 2 | DS1 | DLS | |

Moderate | Moderate damage | Life safety | Moderate | Grade 3 | DS2 | LLS |

Extensive | Extensive damage | Heavy | ||||

Partial collapse | Near collapse | Major | Grade 4 | DS3 | CLS | |

Collapse | Collapse | DS4 |

**Table 3.**Global geometry parameters of the frame typologies in Figure 4.

2-Bay | 3-Bay | |||||
---|---|---|---|---|---|---|

Type 1 | Type 2 | Type 3 | Type 4 | Type 5 | ||

L = 8 m | L_{1}/L | 0.50 | 0.30 | |||

L_{2}/L | 0.50 | 0.70 | ||||

L = 10 m | L_{1}/L | 0.33 | 0.25 | 0.25 | ||

L_{2}/L | 0.33 | 0.50 | 0.25 | |||

L_{3}/L | 0.33 | 0.25 | 0.50 | |||

L = 12 m | L_{1}/L | 0.33 | 0.25 | 0.25 | ||

L_{2}/L | 0.33 | 0.50 | 0.25 | |||

L_{3}/L | 0.33 | 0.25 | 0.50 |

**Table 4.**Ranges adopted for material properties and reinforcement for the two considered sub-typologies.

Median Concrete Strength
${\mathit{f}}_{\mathit{y}\mathit{m}}$ (MPa) | Median Steel Strength
${\mathit{f}}_{\mathit{y}\mathit{m}}$ (MPa) | Stirrup Diameter
${\mathit{\varphi}}_{\mathit{s}\mathit{t}}$ (mm) | Stirrup Spacing
${\mathit{s}}_{\mathit{s}\mathit{t}}$ (mm) | Flexural Reinforcement
${\mathit{A}}_{\mathit{s}}$ (%) | |
---|---|---|---|---|---|

Code-Based (New, 1991–2000) | 18–28 | 300–500 | 8–10 | 150–250 | 0.75–1.25 |

Pre-Code (Old, 1961–1970) | 14–20 | 220–370 | 6–8 | 200–300 | 0.65–1.00 |

Uniform distribution | Uniform distribution | Discrete distribution | Discrete distribution | Discrete distribution |

**Table 5.**Fragility curve parameters (mean ${\mu}_{G}$, fuse width around the mean, and coefficient of variation ${V}_{G}$ of the PGA-based capacity) for three Damage/Limit States, for three soil types (A, B, C), two construction types (“Old”, Gravity Load Design, and “New”, code-based design) and with number of storeys ranging from 1 to 5.

Soil Type | Design Type | Storey | DS1 = DLS | DS2 = LLS | DS3 = CLS | ||||||
---|---|---|---|---|---|---|---|---|---|---|---|

${\mathit{\mu}}_{\mathit{G}}$ | ± | ${\mathit{V}}_{\mathit{G}}$ | ${\mathit{\mu}}_{\mathit{G}}$ | ± | ${\mathit{V}}_{\mathit{G}}$ | ${\mathit{\mu}}_{\mathit{G}}$ | ± | ${\mathit{V}}_{\mathit{G}}$ | |||

Soil A | Old (GLD) | 1 | 0.207 | 0.016 | 0.337 | 0.813 | 0.050 | 0.320 | 1.162 | 0.048 | 0.368 |

2 | 0.158 | 0.022 | 0.332 | 0.478 | 0.048 | 0.323 | 0.902 | 0.052 | 0.356 | ||

3 | 0.152 | 0.018 | 0.336 | 0.380 | 0.052 | 0.316 | 0.817 | 0.048 | 0.295 | ||

4 | 0.151 | 0.019 | 0.339 | 0.406 | 0.044 | 0.309 | 0.785 | 0.044 | 0.284 | ||

5 | 0.149 | 0.020 | 0.337 | 0.377 | 0.040 | 0.321 | 0.690 | 0.046 | 0.286 | ||

New (Code-Based) | 1 | 0.241 | 0.026 | 0.551 | 0.990 | 0.052 | 0.380 | 1.496 | 0.052 | 0.360 | |

2 | 0.198 | 0.020 | 0.505 | 0.636 | 0.044 | 0.338 | 1.275 | 0.048 | 0.352 | ||

3 | 0.188 | 0.023 | 0.488 | 0.501 | 0.038 | 0.325 | 0.954 | 0.048 | 0.330 | ||

4 | 0.183 | 0.018 | 0.478 | 0.533 | 0.046 | 0.325 | 1.043 | 0.042 | 0.353 | ||

5 | 0.170 | 0.016 | 0.483 | 0.506 | 0.042 | 0.323 | 0.934 | 0.046 | 0.313 | ||

Soil B | Old (GLD) | 1 | 0.149 | 0.024 | 0.303 | 0.602 | 0.044 | 0.320 | 0.972 | 0.045 | 0.350 |

2 | 0.133 | 0.023 | 0.298 | 0.433 | 0.048 | 0.379 | 0.843 | 0.050 | 0.359 | ||

3 | 0.125 | 0.020 | 0.302 | 0.388 | 0.036 | 0.377 | 0.684 | 0.042 | 0.296 | ||

4 | 0.127 | 0.016 | 0.305 | 0.315 | 0.046 | 0.379 | 0.642 | 0.038 | 0.299 | ||

5 | 0.124 | 0.018 | 0.303 | 0.292 | 0.040 | 0.361 | 0.571 | 0.044 | 0.266 | ||

New (Code-Based) | 1 | 0.169 | 0.025 | 0.459 | 0.767 | 0.046 | 0.347 | 1.158 | 0.048 | 0.398 | |

2 | 0.135 | 0.021 | 0.421 | 0.516 | 0.035 | 0.360 | 0.960 | 0.047 | 0.312 | ||

3 | 0.139 | 0.018 | 0.406 | 0.460 | 0.040 | 0.422 | 0.871 | 0.044 | 0.300 | ||

4 | 0.134 | 0.022 | 0.398 | 0.432 | 0.038 | 0.410 | 0.821 | 0.040 | 0.284 | ||

5 | 0.127 | 0.019 | 0.403 | 0.406 | 0.036 | 0.428 | 0.750 | 0.050 | 0.260 | ||

Soil C | Old (GLD) | 1 | 0.122 | 0.024 | 0.303 | 0.308 | 0.044 | 0.330 | 0.850 | 0.038 | 0.360 |

2 | 0.088 | 0.020 | 0.298 | 0.300 | 0.036 | 0.337 | 0.747 | 0.042 | 0.332 | ||

3 | 0.074 | 0.018 | 0.302 | 0.244 | 0.042 | 0.281 | 0.596 | 0.044 | 0.289 | ||

4 | 0.069 | 0.016 | 0.305 | 0.209 | 0.034 | 0.268 | 0.516 | 0.048 | 0.249 | ||

5 | 0.064 | 0.018 | 0.303 | 0.199 | 0.040 | 0.242 | 0.474 | 0.050 | 0.243 | ||

New (Code-Based) | 1 | 0.121 | 0.020 | 0.459 | 0.591 | 0.042 | 0.340 | 0.932 | 0.044 | 0.370 | |

2 | 0.096 | 0.018 | 0.421 | 0.409 | 0.038 | 0.320 | 0.868 | 0.052 | 0.323 | ||

3 | 0.090 | 0.022 | 0.406 | 0.375 | 0.040 | 0.333 | 0.795 | 0.048 | 0.298 | ||

4 | 0.088 | 0.017 | 0.398 | 0.302 | 0.046 | 0.361 | 0.707 | 0.044 | 0.289 | ||

5 | 0.085 | 0.018 | 0.403 | 0.280 | 0.048 | 0.311 | 0.671 | 0.048 | 0.282 |

**Table 6.**Fragility curve parameters including soil type (mean ${\mu}_{SG}$, fuse width around the mean, and coefficient of variation ${V}_{SG}$ of the PGA-based capacity), for three Damage/Limit States, two construction types (“Old”, Gravity Load Design, and “New”, code-based design) and with number of storeys ranging from 1 to 5.

Design Type | # Storey | DS1 = DLS | DS2 = LLS | DS3 = CLS | ||||||
---|---|---|---|---|---|---|---|---|---|---|

${\mathit{\mu}}_{\mathit{S}\mathit{G}}$ | ± | ${\mathit{V}}_{\mathit{S}\mathit{G}}$ | ${\mathit{\mu}}_{\mathit{S}\mathit{G}}$ | ± | ${\mathit{V}}_{\mathit{S}\mathit{G}}$ | ${\mathit{\mu}}_{\mathit{S}\mathit{G}}$ | ± | ${\mathit{V}}_{\mathit{S}\mathit{G}}$ | ||

Old (GLD) | 1 | 0.161 | 0.063 | 0.314 | 0.564 | 0.300 | 0.323 | 1.011 | 0.199 | 0.359 |

2 | 0.124 | 0.056 | 0.309 | 0.395 | 0.131 | 0.346 | 0.830 | 0.125 | 0.349 | |

3 | 0.113 | 0.057 | 0.313 | 0.317 | 0.115 | 0.325 | 0.709 | 0.157 | 0.293 | |

4 | 0.112 | 0.059 | 0.316 | 0.313 | 0.138 | 0.319 | 0.649 | 0.181 | 0.277 | |

5 | 0.108 | 0.062 | 0.314 | 0.288 | 0.129 | 0.308 | 0.580 | 0.156 | 0.265 | |

New (Code-Based) | 1 | 0.184 | 0.083 | 0.490 | 0.796 | 0.247 | 0.356 | 1.218 | 0.330 | 0.376 |

2 | 0.148 | 0.070 | 0.449 | 0.526 | 0.155 | 0.339 | 1.070 | 0.254 | 0.329 | |

3 | 0.140 | 0.072 | 0.433 | 0.437 | 0.102 | 0.360 | 0.875 | 0.128 | 0.309 | |

4 | 0.136 | 0.065 | 0.425 | 0.418 | 0.162 | 0.365 | 0.874 | 0.211 | 0.309 | |

5 | 0.127 | 0.060 | 0.430 | 0.390 | 0.158 | 0.354 | 0.802 | 0.179 | 0.285 |

**Table 7.**Fragility curve parameters including soil type and demand variability (mean ${\mu}_{SG}$, fuse width around the mean, and total coefficient of variation ${V}_{T}$ ), for three Damage/Limit States, two construction types (“Old”, Gravity Load Design, and “New”, code-based design) and with number of storeys ranging from 1 to 5.

Design Type | # Storey | DS1 = DLS | DS2 = LLS | DS3 = CLS | ||||||
---|---|---|---|---|---|---|---|---|---|---|

${\mathit{\mu}}_{\mathit{S}\mathit{G}}$ | ± | ${\mathit{V}}_{\mathit{T}}$ | ${\mathit{\mu}}_{\mathit{S}\mathit{G}}$ | ± | ${\mathit{V}}_{\mathit{T}}$ | ${\mathit{\mu}}_{\mathit{S}\mathit{G}}$ | ± | ${\mathit{V}}_{\mathit{T}}$ | ||

Old (GLD) | 1 | 0.161 | 0.063 | 0.373 | 0.564 | 0.300 | 0.514 | 1.011 | 0.199 | 0.538 |

2 | 0.124 | 0.056 | 0.368 | 0.395 | 0.131 | 0.529 | 0.830 | 0.125 | 0.531 | |

3 | 0.113 | 0.057 | 0.372 | 0.317 | 0.115 | 0.515 | 0.709 | 0.157 | 0.496 | |

4 | 0.112 | 0.059 | 0.374 | 0.313 | 0.138 | 0.511 | 0.649 | 0.181 | 0.487 | |

5 | 0.108 | 0.062 | 0.373 | 0.288 | 0.129 | 0.505 | 0.580 | 0.156 | 0.480 | |

New (Code-Based) | 1 | 0.184 | 0.083 | 0.529 | 0.796 | 0.247 | 0.535 | 1.218 | 0.330 | 0.549 |

2 | 0.148 | 0.070 | 0.492 | 0.526 | 0.155 | 0.525 | 1.070 | 0.254 | 0.518 | |

3 | 0.140 | 0.072 | 0.477 | 0.437 | 0.102 | 0.538 | 0.875 | 0.128 | 0.506 | |

4 | 0.136 | 0.065 | 0.469 | 0.418 | 0.162 | 0.542 | 0.874 | 0.211 | 0.505 | |

5 | 0.127 | 0.060 | 0.474 | 0.390 | 0.158 | 0.534 | 0.802 | 0.179 | 0.491 |

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**MDPI and ACS Style**

Rahmat Rabi, R.; Bianco, V.; Monti, G.
Mechanical-Analytical Soil-Dependent Fragility Curves of Existing RC Frames with Column-Driven Failures. *Buildings* **2021**, *11*, 278.
https://doi.org/10.3390/buildings11070278

**AMA Style**

Rahmat Rabi R, Bianco V, Monti G.
Mechanical-Analytical Soil-Dependent Fragility Curves of Existing RC Frames with Column-Driven Failures. *Buildings*. 2021; 11(7):278.
https://doi.org/10.3390/buildings11070278

**Chicago/Turabian Style**

Rahmat Rabi, Raihan, Vincenzo Bianco, and Giorgio Monti.
2021. "Mechanical-Analytical Soil-Dependent Fragility Curves of Existing RC Frames with Column-Driven Failures" *Buildings* 11, no. 7: 278.
https://doi.org/10.3390/buildings11070278